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5 Loop Quantization of Symmetric Models

Analogies prove nothing, but they can make one feel more at home.

Sigmund Freud

Introductory Lectures on Psychoanalysis

In full loop quantum gravity, the quantum representation is crucial for the foundation of the theory. The guiding theme there is background independence, which requires one to smear the basic fields in a particular manner to holonomies and fluxes. In this section, we will see what implications this has for composite operators and the physical effects they entail. We will base this analysis on symmetric models in order to be able to perform explicit calculations.

Symmetries are usually introduced in order to simplify calculations or make them possible in the first place. However, symmetries can sometimes also lead to complications in conceptual questions, if the additional structure they provide is not fully taken into account. In the present context, it is important to realize that the action of a symmetry group on a space manifold provides a partial background such that the situation is always slightly different from the full theory. If the symmetry is strong, such as in homogeneous models, other representations such as the Wheeler–DeWitt representation can be possible even though the fact that a background has been used may not be obvious. While large-scale physics is not very sensitive to the representation used, it becomes very important on the smallest scales, which we have to take into account when the singularity issue is considered.

Instead of looking only at one symmetric model, where one may have different choices for the basic representation, one should keep the full view of different models as well as the full theory. In fact, in loop quantum gravity it is possible to relate models and the full theory such that symmetric states and basic operators, and thus the representation, can be derived from the unique background-independent cyclic representation of the full theory. We will describe this in detail in Section 7, after having discussed the construction of quantum models in the present section. Without making use of the relation to the full theory, one can construct models by analogy. This means that quantization steps are modeled on those, which are known to be crucial in the full theory, which start with the basic representation and continue to the Hamiltonian constraint operator. One can then disentangle the places where additional input, in comparison to the full theory, is needed and what implications it may have.

 5.1 Symmetries and backgrounds
 5.2 Isotropy
 5.3 Isotropy: Matter Hamiltonian
 5.4 Isotropy: Hamiltonian constraint
 5.5 Dynamical refinements of the discreteness scale
 5.6 Semiclassical limit and correction terms
  5.6.1 WKB approximation
  5.6.2 Effective formulation
 5.7 Homogeneity
 5.8 Diagonalization
 5.9 Homogeneity: Dynamics
 5.10 Inhomogeneous models
 5.11 Einstein–Rosen waves
  5.11.1 Canonical variables
  5.11.2 Representation
 5.12 Spherical symmetry
 5.13 Loop inspired quantum cosmology
 5.14 Dynamics
 5.15 Dynamics: General construction
 5.16 Singularities
 5.17 Initial/boundary value problems
 5.18 Pre-classicality and boundedness
 5.19 Dynamical initial conditions
 5.20 Numerical and mathematical quantum cosmology
 5.21 Summary

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